Problem: Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$
Solution: Let $A = (3,-5),$ $B = (-2,0),$ and $C = (1,-6).$  Let $\mathbf{v} = \overrightarrow{CA} = \begin{pmatrix} 3 - 1 \\  -5 - (-6) \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \overrightarrow{CB} = \begin{pmatrix} -2 - 1 \\ 0 - (-6) \end{pmatrix} = \begin{pmatrix} -3 \\ 6 \end{pmatrix}.$  The area of triangle $ABC$ is half the area of the parallelogram determined by $\mathbf{v}$ and $\mathbf{w}.$

[asy]
unitsize(0.6 cm);

pair A, B, C;

A = (3,-5);
B = (-2,0);
C = (1,-6);

draw(A--B);
draw(C--A,Arrow(6));
draw(C--B,Arrow(6));
draw(A--(A + B - C)--B,dashed);

label("$\mathbf{v}$", (A + C)/2, SE);
label("$\mathbf{w}$", (B + C)/2, SW);
dot("$A$", A, E);
dot("$B$", B, W);
dot("$C$", C, S);
[/asy]

The area of the parallelogram determined by $\mathbf{v}$ and $\mathbf{w}$ is
\[|(2)(6) - (-3)(1)| = 15,\]so the area of triangle $ABC$ is $\boxed{\frac{15}{2}}.$